In general, a gauge transformation is defined as a transformation
\(\delta F = \delta \epsilon [F,{\cal G}]\)  that does not alter the physical state of the system. The function \({\cal G}\)  is the gauge generator. We have seen that the primary first class constraints generate gauge transformations. But not all gauge transformations are generated by primary first class constraints. In fact, it can be shown [1]}, [2]} that the Poisson brackets between any primary first class constraint and the first class Hamiltonian is itself a first class constraint that generates a gauge transformation.For systems with more than one primary first class constraint, the Poisson
brackets of any two primary first class constraints is also a first class constraint that generates a gauge transformation [1]}, [2]}.
